Counterexamples to Jacobian Conjecture not surjective

I notice that the recently-publicized counterexample(s) to the Jacobian Conjecture are not surjective. Is that necessarily the case? That is, (Q) If $F:\mathbb C^n \to\mathbb C^n$ is algebraic and everywhere locally injective, and also surjective, must it be injective? Maybe the relevant setting is something broader; is the statement even true if we replace "algebraic" with "differentiable"? (Perhaps now adding the condition that the Jacobian determinant be constant, which is automatic in the algebraic setting but not the differentiable one.)